Lecture Notes: Algorithmic Game Theory Palash Dey Indian Institute of Technology, Kharagpur [email protected] Copyright c 2019 Palash Dey. This work is licensed under a Creative Commons License (http://creativecommons.org/licenses/by-nc-sa/4.0/). Free distribution is strongly encouraged; commercial distribution is expressly forbidden. See https://cse.iitkgp.ac.in/~palash/ for the most recent revision. Statutory warning: This is a draft version and may contain errors. If you find any error, please send an email to the author. 2 Contents I Game Theory7 1 Introduction to Non-cooperative Game Theory9 1.1 Normal Form Game.......................................... 10 1.2 Big Assumptions of Game Theory.................................. 11 1.2.1 Utility............................................. 11 1.2.2 Rationality (aka Selfishness)................................. 11 1.2.3 Intelligence.......................................... 11 1.2.4 Common Knowledge..................................... 11 1.3 Examples of Normal Form Games.................................. 12 2 Solution Concepts of Non-cooperative Game Theory 15 2.1 Dominant Strategy Equilibrium................................... 15 2.2 Nash Equilibrium........................................... 17 3 Matrix Games 21 3.1 Security: the Maxmin Concept.................................... 21 3.2 Minimax Theorem.......................................... 23 3.3 Application of Matrix Games: Yao’s Lemma............................. 27 3.3.1 View of Randomized Algorithm as Distribution over Deterministic Algorithms..... 27 3.3.2 Yao’s Lemma......................................... 28 4 Computing Equilibrium 29 4.1 Computing MSNEs of Bimatrix Games by Enumerating Support................. 29 4.2 Important Classes of Succinct Games................................ 30 4.3 Potential Games............................................ 31 4.4 Approximate PSNE.......................................... 33 4.5 Local Search Problem........................................ 35 4.6 Complexity Class: Polynomial Local Search (PLS)......................... 35 4.7 PLS-Completeness.......................................... 36 4.8 PSNE Computation in Congestion Games.............................. 36 4.9 Complexity Class: Functional NP .................................. 38 4.10 Complexity Class TFNP: NP Search Problems with Guaranteed Witness............. 40 4.11 Complexity Class PPAD: A Syntactic Subclass of TFNP ...................... 40 4.12 A Canonical PPAD-complete Problem: Sperner’s Lemma..................... 41 3 4.13 Algorithms for "-MSNE........................................ 42 1 4.13.1 A Simple Polynomial Time Algorithm for 2 Approximate MSNE............. 42 5 Correlated and Coarse Correlated Equilibrium 45 5.1 Correlated Equilibrium........................................ 45 5.2 Coarse Correlated Equilibrium (CCE)................................ 46 5.3 External Regret Framework..................................... 47 5.4 No-Regret Algorithm......................................... 48 5.5 Analysis of Multiplicative Weight Algorithm............................ 49 5.6 No-Regret Dynamic.......................................... 51 5.7 Swap Regret.............................................. 52 5.8 Black Box Reduction from Swap Regret to External Regret.................... 53 6 Price of Anarchy 57 6.1 Braess’s Paradox........................................... 57 6.2 Pigou’s Network........................................... 57 6.3 PoA for Selfish Networks....................................... 58 6.4 Selfish Load Balancing........................................ 60 6.5 Selfish Load Balancing Game.................................... 61 6.6 Price of Anarchy of Selfish Load Balancing............................. 61 7 Other Forms of Games 63 7.1 Bayesian Games............................................ 63 7.2 Selten Game.............................................. 64 7.3 Extensive Form Games........................................ 66 II Mechanism Design 69 8 Mechanism, Social Choice Function and Its Implementation 71 8.1 Bayesian Game Induced by a Mechanism.............................. 73 8.2 Implementation of Social Choice Function............................. 73 8.3 Revelation Principle......................................... 74 8.4 Properties of Social Choice Function................................ 75 8.4.1 Ex-Post Efficiency or Pareto Optimality........................... 75 8.4.2 Non-Dictatorship....................................... 75 8.4.3 Individual Rationality.................................... 76 9 Gibbard-Satterwaite Theorem 77 9.1 Statement and Proof of Gibbard-Satterwaite Theorem....................... 77 9.2 Way-outs from GS Impossibility................................... 81 10 Mechanism Design in Quasi-linear Environment 83 10.1 Allocative Efficiency (AE)...................................... 83 10.2 Budget Balance (BB)......................................... 84 4 10.3 Groves Mechanism.......................................... 85 10.4 Clarke (Pivotal) Mechanism..................................... 86 10.5 Examples of VCG Mechanisms.................................... 87 10.6 Weighted VCG............................................ 89 10.7 Single Parameter Domain...................................... 91 10.8 Implementability in Intermediate Domains............................. 94 10.9 A Glimpse of Algorithmic Mechanism Design: Knapsack Allocation............... 95 11 Mechanism Design Without Money 97 11.1 Stable Matching............................................ 97 11.2 House Allocation........................................... 101 Notation: N = f0, 1, 2, ...g denotes the set of natural numbers, R denotes the set of real numbers. For a set X, its power set is denoted by 2X. 5 6 Part I Game Theory 7 Chapter 1 Introduction to Non-cooperative Game Theory Game theory is a field in mathematics which studies “games.” Intuitively speaking, a game is any “system” where there are multiple parties (called players of the game), the “outcome” depends on the actions that individual parties perform, and different parties derive different utility from the outcome. Let us see some examples of games to have a better feel for the subject. Our first example is arguably the most exciting one from the students’ perspective. Example 1.0.1 (Grading Game). Consider the “Algorithmic Game Theory” class in IIT Kharagpur in 2019. Suppose the instructor announces that the grading policy will be as follows – the top 10% of students get EX grade, next 20% get A, etc. Could you see the game that this grading policy induces? The players are the students in the class. The outcome of the system is the function which maps students to the grades that they get. Observe that the outcome of the system depends on the actions of all the players. Our next example is the famous prisoner’s dilemma. Example 1.0.2 (Prisoner’s Dilemma). Consider two persons have committed a crime and subsequently been caught by the police. However, the police do not have enough evidence to make a strong case against them. The police ask them separately about the truth. Both the prisoners each have the following options – either tell the truth that they have committed the crime (let us call this action C) or lie that they have not committed the crime (let us call this action NC). The prisoners also know the following – if both the prisoners play C, then police has a strong case and both the prisoners get 5 years of jail terms; if both the prisoners play NC, then the police do not have a strong case and both the prisoners get 2 years of jail terms; if one prisoner plays C and the other plays NC, then the police again have a strong case and the prisoner who played C gets 1 year of jail term (less punishment for cooperating with the police) whereas the other prisoners get 10 years of jail terms. 9 Our next example is called congestion games which model the congestion on road networks, Internet traffic, etc. Example 1.0.3 (Congestion Games). Suppose every people in a city want to go from some point A in the city to some other point B and each person decide the path that they wish to take. The speed with which people can commute on any road is inversely proportional to the number of people using that road. We can again observe that the situation again induces a game with commuters being its players and the time that every commuter takes to reach its destination being the outcome. The above way of modeling situations in terms of games is so fundamental that we can keep on giving more and more examples of games from almost every field that we can think about. We now formally define games. The most common form of games is the normal form games aka strategic form games. 1.1 Normal Form Game Normal Form Game Definition 1.1. A normal form game Γ is defined as a tuple hN, (Si)i2N, (ui)i2Ni where B N is the set of players B Si is the set of strategies for the player i 2 N B ui : ×i2NSi −! R is the utility function for the player i 2 N Let us write the prisoners’ dilemma game in the normal form. Example 1.1.1 (Prisoners’ Dilemma in Normal Form). The prisoners’ dilemma game in the normal form can be written as follows. B The set of players (N): f1, 2g B The set of strategies: Si = fC, NCg for every i 2 [2] Player 2 C NC B Payoff matrix: C (-5, -5) (-1, -10) Player 1 NC (-10, -1) (-2, -2) Okay, suppose we have modeled some scenario as a game. Then what? What are the questions we would like to answer? One of the most important questions that we would like to